= Solution
Let $V=L^3$ be the box <volume>. The <number operator> is the sum of the <occupation numbers> of all <momentum eigenstates>, including the zero mode:
$$
\boxed{\hat N=\int_V d^3r\,\psi^\dagger(\mathbf r)\psi(\mathbf r)
=\sum_{\mathbf k}\psi_{\mathbf k}^\dagger\psi_{\mathbf k}.}
$$
In the <Fourier series> for the <contact interaction>, the two <creation operators> supply phase $e^{-i(\mathbf k_1+\mathbf k_2)\cdot\mathbf r}$ and the two <annihilation operators> supply $e^{i(\mathbf k_3+\mathbf k_4)\cdot\mathbf r}$. <Fourier orthogonality> with <periodic boundary conditions> gives
$$
\int_V d^3r\,e^{i(\mathbf k_3+\mathbf k_4-\mathbf k_1-\mathbf k_2)\cdot\mathbf r}
=V\delta_{\mathbf k_1+\mathbf k_2,\,\mathbf k_3+\mathbf k_4}.
$$
The four normalization factors together contribute $V^{-2}$, leaving the <momentum-space contact-interaction Hamiltonian>
$$
\boxed{H_{\rm int}=\frac{u}{2V}
\sum_{\mathbf k_1,\mathbf k_2,\mathbf k_3,\mathbf k_4}
\delta_{\mathbf k_1+\mathbf k_2,\,\mathbf k_3+\mathbf k_4}
\psi_{\mathbf k_1}^\dagger\psi_{\mathbf k_2}^\dagger
\psi_{\mathbf k_3}\psi_{\mathbf k_4}.}
$$
The <Kronecker delta> enforces <momentum conservation>. An equivalent expression that resolves the momentum transfer is
$$
H_{\rm int}=\frac{u}{2V}\sum_{\mathbf k,\mathbf p,\mathbf q}
\psi_{\mathbf k+\mathbf q}^\dagger\psi_{\mathbf p-\mathbf q}^\dagger
\psi_{\mathbf p}\psi_{\mathbf k}.
$$
Both are already in <normal ordering>; no extra <commutator> constant is introduced.
Back to article page
